Mathematicians who work in mathematical analysis - particularly what is called classical real or complex analysis - don't often say such-and-such is infinite, rather they say a process tends to infinity, provided, for every bound one might conjecture, the process eventually moves beyond that bound. For example, suppose we state

$\underset{x\to a}{\mathop{Lim}}\,F(x)=\infty$

We may say, "F(x) goes to infinity as x goes to a", but what we mean is that for each
$M\gt0$ there exists a positive number $\varepsilon =\varepsilon (M)$
such that

$\left| x-a \right|\lt\varepsilon \Rightarrow F(x)\gt M$


$-\infty$ can be used in a similar manner. And there are comparable definitions for complex valued functions in the complex plane.

It doesn't go much beyond this sort of (Cauchy-Weierstrass) definition in analysis. Orders of infinity and the like normally don't appear in the literature. However, soft or modern analysis does move in the general direction of set theory and algebra. And set theory is a different story; most of the posts in these kinds of threads pertain to that subject. :cool:

2+1=3 is contingently true; not necessarily true. But don't fret. The entire thread is unnecessary babble.

:roll:

↪Pfhorrest It is still producing no mirth in me. — Bartricks

To be mirthless at such wit means you are behind the times. :smirk:

Physics tells us that even bricks are nothing more than probability waves smeared across the universe — fishfry

Aha! So, finally, a resolution to the question I posed about PWs on another thread. The medium through which they travel are brick roads. And they culminate on the shores of the Emerald City!

Thanks! :nerd:

Analytic philosophy is just industrialized thinking. Robots will do it better in a few years. — Pneumenon

Perhaps the same will be said of mathematics. There's a crew working indirectly on that project on this very forum. :smile:

I still like the idea of a metaphysical medium through which probability waves travel.

But, assuming the PW is itself a metaphysical entity, then it is a metaphysical actuality since it "describes" a physical observation. A previous thread mentioned this notion, and I find it entertaining to contemplate. :cool:

2+1=1 mod(2)

Thus your entire argument falls apart. Sorry :cry:

Yes, there is a wave function, and yes, the function is probabilistic, but that doesn't make the function a 'wave of probability' traveling through a medium — noAxioms

"By analogy with waves such as those of sound, a wave function, designated by the Greek letter psi, Ψ, may be thought of as an expression for the amplitude of the particle wave (or de Broglie wave), although for such waves amplitude has no physical significance." Encyl. Brit.

I was suggesting a metaphysical argument. Interested in seeing what it would provoke. :snicker:

QM does not posit waves of probability, through a medium or otherwise. — noAxioms

Of course it does, but not in a physical sense.

From Physics.Org: "At the heart of quantum mechanics lies the wave function, a probability function used by physicists to understand the nanoscale world. Using the wave function, physicists can calculate a system's future behavior, but only with a certain probability. This inherently probabilistic nature of quantum theory differs from the certainty with which scientists can describe the classical world, leading to a nearly century-long debate on how to interpret the wave function: does it representative objective reality or merely the subjective knowledge of an observer?"

What about thinking of tangent vectors to a circle in the complex plane rather than a sphere? But I'm not following closely. :sad:

The Wiki article on Analytic Philosophy is a shocker — Banno

I'm no philosopher, but the article looks good to me. What in particular do you find in error? Be specific, please. :chin:

I still haven't come to a conclusion regarding metaphysics in the realms of science and mathematics. At times I feel that only scientists who are involved in or knowledgeable of research into a particular topic are qualified to delve into the metaphysics of that topic, and at other times I think that this requirement is too stringent.

What do others think? :chin:

What is the medium through which probability waves in QM travel?

How about it, physicist out there? Clarify the idea that MU advances? Waves in fields create particles? Good luck with the metaphysics of fields. :nerd:

One senior level college course, taken before the time of the dinosaurs.

A PhD in math, during the time of the dinosaurs.

Out to pasture, now. :cool:

Empirical evidence indicates that there are waves and this necessitates the conclusion of an "ether" or some such substance which the waves exist in — Metaphysician Undercover

I would like to hear this from a physicist. :roll:

I can summarize. Short answer is that these days you can do logic via category theory; and when you do that, you get intuitionist logic (denial of the law of the excluded middle (LEM) and all that) in a natural way. — fishfry

I really appreciate your explanation. Thank you. I looked briefly at your first link to get an idea of the univalent approach. I was completely unaware of this, being happily non-constructive at times! :smile:

I said, in time Wikipedia will be cited, because I think there will be significant changes to academic vetting. Not now. I never had any problems publishing articles in math journals, but when you get a chance to look under the surface in a discipline you see the problems - and the profit motives. The scholar does the work, the reviewers do the work, and the institution pays the publisher, then other institutions pay for subscriptions. Nice racket. Has that changed? I know all the reasons for the system, but it is mildly corrupt.

On the other hand the proliferation of online journals with weak refereeing standards exacerbates problems.

I apologize for interrupting a productive flow of thought. But I was curious what you guys were talking about. Seems pretty esoteric. :chin:

"Would any expert academic in your field cite a Wikipedia page in a peer review article?"

Most of my publishing was done years ago, before Wikipedia, and I don't read the journals very much anymore. So I can't say with any degree of accuracy. But times are rapidly changing, and in the past I have found mistakes in reviewed articles, so I would guess that in time Wikipedia will be cited if merely to acquaint a reviewer with arcane material or the latest breakthroughs.

The entire structure of reviewing, refereeing, and dissemination should change and should put journal publishers out of business. This should be an age of open, free discussions. :chin:

for reasons is always true — TheWillowOfDarkness

In ancient Greece a bolt of lightning was reasoned to be due to Zeus. Reason evolves.

Why would an expert write a Wikipedia page? — Bartricks

Mathematics is a social or community endeavor to a large extent. And, yes, many experts from that community pay attention to what is written on their areas' Wiki pages, correcting mistakes and contributing info. The only other subject I'm familiar with is climbing, and, there, things are not quite as disciplined, but still mostly accurate.

Several years ago there was an effort to compare the accuracy of articles on the same subject appearing in Encyclopedia Britannica(online) and Wikipedia. If I recall correctly, in general Wikipedia was slightly more accurate than EB.

Secondly, the only subject that studies patterns seriously is math and I don't know if there's a subfield devoted to just the study of patterns or not. — TheMadFool

https://www.maa.org/press/periodicals/convergence/mathematics-as-the-science-of-patterns-mathematics-as-the-science-of-patterns :cool:

You would still have to follow the regulations of the wikipedia regulatory framework. Do you know its rules and how they are enforced? If not, then you are yourself not an expert on wikipedia. — alcontali

I consider wikipedia a very good source for introductory material on many math topics. If a topic is fairly popular, it is likely to be accurate. For minor topics not quite so accurate at times. The same can be said of peer reviewed articles, having been there and done that while active. :cool:

analytical continuation of a complex function has something to do with chaotic systems? Did I understand correctly? — Mephist

Dynamical systems! Actually, I was more general, showing certain sequences of linear fractional transformations can have their regions of convergence expanded by the use of fixed points. Continued fractions are a special case.

Here is a simple example illustrating the use of a fixed point:

Given $F(z)=\frac{1}{1-z}$

Then $F(z)=1+z+{{z}^{2}}+\cdots ,\text{ }\left| z \right|\lt1$

For $\beta=0$ the T-fraction expansion of the power series is

${{T}_{n}}(\beta )=1+\frac{z}{1-2z+\frac{z}{1-z+\frac{z}{1-z+\ldots +\frac{z}{1-z+\beta }}}}$

The continued fraction value is by definition

${{T}_{n}}(0)\to CF$


Here it is found that

${{T}_{n}}(0)\to F(z),\text{ }\left| z \right|\lt1$ and

${{T}_{n}}(0)\to \tfrac{1}{2},\text{ }\left| z \right|\gt1$

Whereas

${{T}_{n}}(\beta )\to F(z),\text{ }\left| z \right|\gt1,\text{ }\beta =z$

Here
$\beta =z$ is the repelling fixed point of the function $t(\zeta)=\frac{z}{1-z+\zeta }$

Someone with credentials would or would at least know what to do to test whether we've got the real-deal on our hands — Bartricks

Makes me regret not having those mysterious credentials you may or may not have that would allow you and I to debate issues that may or may not resonate with those who consider themselves authorities, but may not be. I feel defective. :cry:

I had one semester of senior-level philosophy. You?

My prof laughed at metaphysics. Although I proposed Leibniz's monads to him as a legitimate metaphysical actuality. I think it is. :smirk:

But l think there are artists who don't want their art to be public knowledge and they keep it to themselves. — Wittgenstein

And doing so, make it difficult to ascertain if they exist. But, assuming they do exist, one cannot determine whether their virtual art is really art: did they have the intention to produce art, which may or may not exist? :chin:

Do you have any formal qualifications in philosophy? — Bartricks

Please display yours. If they exist I'm not seeing much evidence. :gasp:

But then, I have almost none, so what do I know? (I can guess your reply) :sad:

Well, not only is useful, but if you find a relation between apparently completely different areas of mathematics, maybe those concepts have in some way a deeper meaning. — Mephist

Yes, I'm familiar with the notion, although I have no experience in an algebraic venue. For instance, many years ago I showed that convergence of complex limit periodic continued fractions useful as functional expansions could be accelerated by employing a feature of dynamical systems: attracting fixed points (Proceedings of the AMS). And could be analytically continued by using repelling fixed points (Mathematica Scandinavica and Proc. Royal Norwegian Soc. of Sci. & Letters). There are deeper meanings here by locating these concepts in theory of infinite compositions of complex functions.

I also showed that the traditional Tannery's theorem makes far more sense when embodied in more general infinite compositions rather than merely series and products, or integrals. Not quite what you are stating, but close. You guys are on a roll! :cool:

Yes. I predict, like I say, that you will either end up questioning the probative value of any argument for anything, or you will dismiss the argument on the grounds that it has premises that entail its conclusion (which, I suppose, amounts to the same thing). — Bartricks

Actually, no. Reason is a gift from a god since we are endowed but not capable of conceiving. Sounds a tad like archaic Greek logic. When, according to Jaynes, thoughts were god-given.

But I'm sure I have missed the more subtle features of the argument. Color me embarrassed. :yikes:

Beyond my meager knowledge of topology. I'm more a metric space guy. I hope others reading this material can follow it. Is there value in showing the same algebraic structure? Like then using results in one to prove results in the other?

The specific proof that I was talking about, however, is not currently widely known. So we cannot really look to the expert community's judgement about it, for it has not yet been formed. — Bartricks

Secret knowledge. I look forward to the revelation, although I may be ill-equipped to critique it. :chin:

Do you see how that just takes us back to the beginning.” I don’t know what art is, but I know what I like.” — Brett

Sure. It's a product of the artist's conscious and subconscious mind which resonates with both my conscious and possibly subconscious mind.

Wiki: "Chaos theory is a branch of mathematics focusing on the study of chaos—states of dynamical systems whose apparently-random states of disorder and irregularities are often governed by deterministic laws that are highly sensitive to initial conditions."

The article goes on to state that it is an interdisciplinary area, however. "Chaotic behavior" might be more accurate outside the mathematical umbrella. Using "theory" makes the concept more formal and technical. Just my two bits. :smile:

But why? — Brett

Because it moves me in a way difficult to describe. I see it and I think,"This is art."

But I see some of the imagery I create with my math and computer programs the same way, even though I have little if any intention of creating art. Merely curiosity, which apparently is the wrong intention, according to Artemis.

I've lost track of this lengthy discussion and how it relates to the topic of the thread. Can one of you describe in layman's terms what you are attempting to do? Even though I'm a retired mathematician it's mostly beyond me.

If a simple explanation is not possible, say so. That's OK :chin:

I don’t think the Nefertiti bust is a work of art in the sense we see it — Brett

I don't think the creator's intention was to create art. The bust was probably created primarily to please the monarchy with a flattering image. This is assuming it is not a fake.

I perceive it as art. :cool:

Is that a statement or question? — Brett

So, if I had thought,"I'm going to do art" the first time and did exactly the same procedure, that first image would have been art? This is a tad more complicated than putting a brush to canvas. In my case the "brush" has a "mind" of its own. — jgill


Right. It's like the difference between accidentally pressing the button on your camera (complicated machine!) and choosing to do so. The camera may be doing much of the "work" (i.e., showing a "mind" of its own), but you're the primary mover.

We have to make that distinction or else you have no way to distinguish art from bird's nests and sunflowers and sunsets. — Artemis

This was part of the conversation about intention being necessary when creating art.

So, from this perspective, how are we to know if the famous bust of Nefertiti is really a work of art? We can't simply gaze at it in admiration, thinking, "What a lovely work of art." What were the intentions of the unknown sculpturer?

I don't agree with this idea.

Bartricks has knowledge about the existence of God that he will not divulge since we are untrained in the intricacies of metaphysical analysis, and thus unworthy. But if we collectively plead with him, he might give us a glimpse of the Truth. :sad:

Is that a statement or question?

There is no concept of accidental art. Accidental art is a moment that happens unexpectedly and the artist is able to use all their skills to take advantage of it. — Brett

If you look at comments on a previous page I indeed posed this question to a member who seems to have some expertise in art. She said that if I produced a pleasing product with no intention of it being art, than, no matter how skillfully done or appealing, it is not art. However, if I were to produce the same product with an intention of creating art, it would be art.

Amazing, huh? :smirk:

But patterns are nothing more than what humans perceive is beautiful, — Gregory

There are patterns that are not beautiful. For instance, an aerial view of a battlefield having a symmetric array of exploded mines. Or a pattern of murders by a serial killer.

But how do we know the pattern is not controlling us? — Gregory

Do you mean that by simply contemplating a pattern it might somehow control us? Or at least influence our thinking? Like mandalas?

Wiki: "In various spiritual traditions, mandalas may be employed for focusing attention of practitioners and adepts, as a spiritual guidance tool, for establishing a sacred space and as an aid to meditation and trance induction."